Following [1]}, [2]} (see also [3]} for the moduli space of spin curves), one can resolve the first problem by allowing for extra rational destabilizing components on stable degenerations of \(X\) . More precisely, we define the compactified universal Jacobian \(\overline{\mathcal {J}}_{g,n}\)  as the algebraic stack parametrizing pairs \((X,L)\)  consisting of a quasi-stable curve \(X\)  of type \((g,n)\) , i.e. an \(n\) -pointed genus-\(g\)  semistable curve whose destabilizing (or exceptional) components are isolated, and a line bundle \(L\)  on \(X\)  which is admissible, i.e. it has degree one on every destabilizing component of \(X\)  (see Definition REF ).
The compactified universal Jacobian \(\overline{\mathcal {J}}_{g,n}\)  decomposes as well into smooth connected components \(\overline{\mathcal {J}}_{g,n}=\coprod _{d\in \mathbb {Z}} \overline{\mathcal {J}}_{g,n,d}\) , according to the degree of the line bundles, each containing \(\mathcal {J}_{g,n,d}\)  as dense and open substack with normal crossing boundary.
Moreover, the compactified universal Jacobian \(\overline{\mathcal {J}}_{g,n}\)  is endowed with a forgetful-stabilization morphism that satisfies the existence part of the valuative criterion for properness (although it is not universally closed since it is not quasi-compact)
\(\begin{aligned}\Phi : \overline{\mathcal {J}}_{g,n} & \longrightarrow \overline{\mathcal {M}}_{g,n}\\(X,L) & \longmapsto X^{\operatorname{st}},\end{aligned}\) 
